- Email: [email protected]
Exotic quark structure of Λ (1405) and scalar nonet in QCD sum rule
ARTICLE IN PRESS
Physica E 40 (2007) 410–413 www.elsevier.com/locate/physe
Exotic quark structure of L(1405) and scalar nonet in QCD sum rule T. Nakamuraa,, J. Sugiyamaa, N. Ishiib, T. Nishikawaa, M. Okaa a
Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan b Center for Computational Science, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan Available online 23 June 2007
Abstract Mixing of multi-quark components in the non-singlet scalar mesons and the flavor singlet L(1405) baryon with negative parity are studied in the QCD sum rule. We propose a formulation to evaluate cross-correlators of q¯q(qqq) and qq¯qq¯ ðqqqq¯qÞ operators and to define mixing of different Fock states in the sum rule. It is applied to the non-singlet scalar mesons and L(1405) baryon. We find that the multi-quark operators predict lower masses than the q¯q(qqq) operators and that the multi-quark states are dominant in the lowest mass states. r 2007 Elsevier B.V. All rights reserved. PACS: 12.38.Lg; 12.39.Mk Keywords: Exotic hadron; QCD sum rule
1. Introduction Recently, the spectroscopy of old (ordinary) hadrons has been re-examined and various possibilities of multiquark components of hadrons have been pointed out. We propose a new analysis of L(1405) and scalar nonet mesons as multi-quark states in QCD sum rule (QCDSR). We, in particular, consider possible mixing of different Fock states. Unnatural spectrum of the light scalar mesons indicates that they are tetra-quark states [1]. The lightest negative-parity baryon L(1405) remains as a mystery in baryon spectroscopy and attracts a lot of attention in the context of strong anti-K-nucleon interactions. We consider here two possible structures of these hadrons: (1) p-wave q¯q or qqq states, and (2) s-wave tetra- or penta-quark states. 2. Formalism In the meson case, the sum rule is obtained by expressing a two point function Z 2 Pðp Þ ¼ i d4 x eipx h0jJðxÞJ y ð0Þj0i, (1)
in two ways. One of them is based on the operator product expansion (OPE), where Eq. (1) is calculated in deep Euclidean region, p2 ¼ 1, and is described in terms of the QCD parameters. The other one is based on a phenomenological parametrization of the spectral function. The spectral function at the physical region (p2 40) is assumed to have a sharp peak resonance at p2 ¼ m2 and continuum at p2 4sth . The sum rule is obtained by matching two expressions, so that the mass of the resonance, m, and the other phenomenological parameters can be expressed by the QCD parameters. Two expressions of the correlators are connected by the dispersion relation. To simplify the sum rule, we approximate the continuum spectrum by the same form as that of the OPE calculation. In order to 2 2 suppress large p2 region by the factor of ep =M , and thus the effects of the continuum, we further apply the Borel transformation. In the baryon case, a similar formalism is used. We employ the following local operators for a 0 and flavor singlet baryon L : J 2 ¼ ð¯ua d a Þ, J 4 ¼ abc dec ðd Ta Cg5 sb Þð¯sd g5 C u¯ Te Þ, J 3 ¼ abc ½ðuTa Cg5 d b Þsc ðuTa Cd b Þg5 sc ðuTa Cg5 gm d b Þgm sc ,
Corresponding author.
E-mail addresses: [email protected], [email protected] (T. Nakamura). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.06.043
J 5 ¼ abc def cfg ðd Ta Cg5 sb ÞðsTd Cg5 ue Þg5 C¯sTg þ ðflavor cyclicÞ,
ð2Þ
ARTICLE IN PRESS T. Nakamura et al. / Physica E 40 (2007) 410–413
411
where a; b; . . . represent colors and C ¼ ig2 g0 . Similarly, the 2-quark and 4-quark operators for K 0 are given by the SU f ð3Þ rotation from the fields for a0 . The OPE is truncated after the dimension-6 terms for the J 2 –J 2 correlator, P22 (or the J 3 –J 3 correlator, P33 ). In order to deal with the same order of power corrections, we expand the cross-correlator, J 2 –J 4 , P24 (P35 ) up to dim-9 and P44 (P55 ) up to dim-12. As usual, high dimension operators are evaluated by the vacuum saturation approximation. We consider the diagrams containing the q¯q annihilation because it is necessary for dealing with the mixing of different Fock states. The quark pair annihilations are substituted for h¯qqi or h¯qgs s Gqi. Because the interpolating fields are under the normal ordering, the perturbative part of the q¯q annihilation must disappear. Because the OPE is represented as a polynomial in x, only the zeroth order term survives in the x ! 0 limit. The main goal of this study is to determine whether the scalar mesons and L are dominated by multi-quark states. It turns out that such a mixing is not easily quantified. It would be natural to consider the strengths of the couplings of the 2-quark (3-quark) and 4-quark (5-quark) operators to the physical state and then evaluate the mixing angle. However, such a procedure is largely dependent on the definition and normalization of the local operators. Indeed, a numerical factor can be easily hidden in the local operators and thus the magnitudes of the coupling strengths are ambiguous. Here, we propose two ways to ‘‘define’’ the ratio of the Fock space probability in the mixing of q¯q and qq¯qq¯ for the scalar (a0 ) meson or similarly for L . In the first approach, we define local operators ‘‘normalized’’ in the context of a full 4-quark operator J 4 in Eq. (2). In fact, the 4-quark operator J 4 contains q¯q component, which is obtained by contracting q¯ q pair into the quark condensate,
Therefore, it does not necessarily have a direct relation to the mixing parameters employed in the quark models. In order to define a mixing angle more appropriately to the quark models, one must determine the normalization of the operators using quark model wave functions. To this end, we employ the bag model and compute the matrix elements,
J 4 ðxÞ ¼ J 04 ðxÞ þ 16h¯ssiJ 2 ðxÞ . |fflfflfflfflfflffl{zfflfflfflfflfflffl}
3. Results and conclusion
(3)
J 02
We regard J 02 and J 04 as ‘‘normalized’’ 2-quark and 4-quark fields, respectively. The quark condensate supplies the dimension of the normalization. Using J 02 ðxÞ and J 04 ðxÞ, we define the mixing parameter, y, so that J a ðxÞ ¼ cos yJ 02 ðxÞ þ sin yJ 04 ðxÞ couples to the physical state most strongly h0jJ a ðxÞja0 i ¼ lfðxÞ,
(4)
where fðxÞ denotes the wave function of the center of mass motion of a0 (i.e., a plane wave for a momentum eigenstate). The mixing parameter can be evaluated from correlation functions with an assumption that the poles are at the same position. This definition of the mixing is model independent, but it depends on the choice of the local operators.
h0jJ 2 ð0Þjð¯qqÞP0 ibag ¼ l2 fð0Þ, h0jJ 4 ð0Þjðqq¯qq¯ ÞS0 ibag ¼ l4 fð0Þ. Now, assuming the bag model states (with definite number of quarks) are normalized properly, one can use l2 and l4 for the normalizations of the operators J 2 and J 4 , respectively. In calculating the mixing parameter, one needs only the ratio of l2 and l4 , which is given by l4 =l2 0:24iR32 =R64 . R is the bag radius and gives the dimensional scale of the normalizations of the two operators that have different dimensions. We assume here that the bag radius of the q¯q state is the same as that of qq¯qq¯ state. Thus, the physical state for a0 is given by the mixing of the two states, ja0 i ¼ i cos yjð¯qqÞP0 i þ sin yjðqq¯qq¯ ÞS0 i ¼ cos yja0 ð2qÞi þ sin yja0 ð4qÞi. Note here that the factor i is necessary to keep the phases of the 2- and 4-quark states in accord so that the mixing parameter can be defined as a real parameter. In the QCDSR, the mixing parameter with the bagmodel normalization can be calculated as R Im d4 x eipx h0jJ 04 ðxÞðJ 04 Þy ð0Þj0i l4 2 2 ¼ (5) R 4 l tan y. 2 Im d x eipx h0jJ 2 ðxÞJ y2 ð0Þj0i This definition is model dependent, but it gives a direct interpretation associated with the quark model.
We calculate the masses of a0 in the case of pure 2-quark and pure 4-quark. The values of QCD parameters are taken as ms ¼ 0:12 GeV, h¯qqi ¼ ð0:23 GeVÞ3 , h¯ssi ¼ 0:8 h¯qqi, h¯qgs s Gqi=h¯qqi ¼ 0:8 GeV2 , has p1 G 2 i ¼ ð0:33 GeVÞ4 and as ¼ 0:4. We find that the Borel stability is fairly good. The positions of the poles in the 2-quark state and the 4-quark states are close to each other. The 4-quark state is slightly light. We plot the mass of the mixed a0 and the mixing parameter defined by the first normalization method in the case of various threshold parameter, sth , in Fig. 1. We see the mixing parameter is almost independent of the threshold and the Borel mass, M. The weak M dependence of the mixing parameter, which is dimensionless, is attributed to cancellation of the p-dependences (M-dependences) of the 2-quark and 4-quark correlators. The mixing parameter is about 701, or in other words, the 4-quark component
ARTICLE IN PRESS T. Nakamura et al. / Physica E 40 (2007) 410–413
occupies 90% of a0 . The predicted mass extracted from the mixed operator, J a ðxÞ, is about 0.9–1.1 GeV, which agrees well with experiment. The mixing parameters from the normalized operators by the bag model are estimated for various bag radii. We take the central value as R ¼ 4:8 GeV1 , which is the bag radius determined for the q¯ q component of a0 . The mixing parameter for R ¼ 5:2 GeV1 is almost the same as the one in the first normalization method. We find that the dominant components of a0 , about more than 70%, are the 4-quark state for R44:4 GeV1 and for the first normalization method. We examine how well these sum rules work. We calculate the pole dominance signature defined by B POPE ðp2 Þyðsth p2 Þ . (6) B½POPE ðp2 Þ We set the Borel window at the region where the pole dominance is more than 30%. This constraint is weaker than the standard criterion, but the results are not sensitive to the choice of this value. One finds that the pole dominance for the pure 4-quark correlator is less than the pure 2-quark correlator. The reason is that the OPE of the 4-quark correlator contains higher powers of p2 and grows rapidly for large p2 . The sum rule for the mixed operator shows similar results. We set the Borel window Mo1:2 GeV and conclude that the 4-quark component is dominant in a0 . The results of K 0 are obtained by the SU f ð3Þ rotation from the results for a0 . Comparing to a0 , the mass extracted from the 2-quark correlator for K 0 is heavier, while that of the 4-quark components lies at about the same mass as a0 . The mass of the experimental values for K 0 has been reported at about 0.84 GeV, and its width seems very wide, about 600 MeV. The scalar nonet 1.4
90.0 (100%)
1.3
67.5 (85%)
1.2
45.0 (50%)
1.1
22.5 (15%)
spectrum from our sum rule for 4-quark components is more consistent with the experimental value than that for 2-quark components. The position of the pole of K 0 for the 2-quark operators starts splitting at low M from that for the 4-quark operators. We, however, evaluate the mixing parameter by assuming that the positions of the poles are at the same position. One finds that the 4-quark component is dominant, occupying about 70–90% of K 0 . These results for K 0 resemble those for a0 . The behavior of the pole dominance for K 0 is also similar to that for a0 . Next we calculate the mass of L by the pure 3-quark and 5-quark operators. It is found that the low-lying spectrum is dominated by the negative parity state. We find the pole positions in 3-quark state and 5-quark state are close. Therefore, we estimate the mass of the mixed L and the mixing parameter defined by the first normalization method in the case of various threshold parameters, sth , in Fig. 2. We see the mixing parameter is about 701, or the 5quark component occupies 90% of L . In the second normalization, we take the central value as R ¼ 5:5 GeV1 , which is the bag radius determined for the qqq component. Then we find that the dominant component of L , about more than 90%, is the 5-quark state. We set the Borel window in the region where the pole dominance is more than 30%. We find the Borel window Mo1:4 GeV and conclude that the 5-quark component is dominant in L . The obtained mass is consistent with the observed baryon, L(1405). We summarize the results in this work. A formulation is proposed to take into account mixing of different Fock states in the QCDSR. We apply the formulation to the scalar mesons and the flavor singlet L . Our sum rules indicate that the 4-quark component occupy 70–90% of the scalar mesons. We find that K 0 is lighter than a0 , which is consistent with the strange-quark counting when these
0.0
(0%)
–22.5 (15%)
0.9 mass (√sth =1.3GeV) mixing parameter (√sth =1.3GeV) mass (√sth =1.4GeV) mixing parameter (√sth =1.4GeV) mass (√sth =1.5GeV) mixing parameter (√sth =1.5GeV)
0.8 0.7 0.6 0.8
1 M [GeV]
–45.0 (50%) –67.5 (85%) –90.0 (100%) 1.2
Fig. 1. The masses of the a0 mixed between 2-quark and 4-quark components are plotted as a function of the Borel mass, M, in the case of pffiffiffiffiffiffi various threshold parameters, sth ¼ 1:3, 1.4 and 1.5 GeV. The mixing parameter of the first normalization method is also plotted and it seems identical in the cases of any threshold parameters.
90.0 (100%) 67.5 (85%)
1.6
45.0 (50%) Mass of Λ* [GeV]
1
mixing parameter [degree]
Mass of a0 [GeV]
1.8
1.4
22.5 (15%)
1.2
0.0
–22.5 (15%)
mass ( √sth =1.6GeV) mixing parameter ( √sth =1.6GeV) mass ( √sth =1.8GeV) mixing parameter ( √sth =1.8GeV) mass ( √sth =2.0GeV) mixing parameter ( √sth =2.0GeV)
1 0.8 0.6 1
1.2
1.4 1.6 M [GeV]
1.8
(0%)
–45.0 (50%)
mixing parameter [degree]
412
–67.5 (85%) –90.0 (100%) 2
Fig. 2. The mass of L , where the 3-quark and 5-quark components are mixed, is plotted as a function of the mass, M, for various threshold pffiffiffiffiffiffi parameters, sth ¼ 1:6, 1.8 and 2.0 GeV. The mixing parameter according to the first normalization method is also plotted. The results for three choices of sth are almost identical.
ARTICLE IN PRESS T. Nakamura et al. / Physica E 40 (2007) 410–413
states are considered as 4-quark states. L(1405) is also dominated by the penta-quark component, whose probability is estimated as more than 90%. There exist several other studies of the 4-quark scalar nonet in the QCDSR. Two of them [2,3] are consistent with our result in terms of reproducing the lighter 4-quark mass than the 2-quark mass. In these analyses, K 0 is predicted to have smaller mass than a0 . We think that the difference of the threshold parameter probably makes K 0 lighter than a0 . Another work [4] claims that no signal is obtained for the 4-quark scalar nonet. We note that their analysis differs from ours in not considering the mixing diagrams, truncating dimension in the OPE, interpolating field and the definition of the coupling strength of ground state. It is important to examine the decay widths and branching ratios of those multi-quark states. The mechan-
413
ism for the fall-apart decay in which the multi-quark hadrons dissociate into two color-singlet hadrons without creating q¯q pairs. The widths associated with the fall-apart processes depend strongly on the quantum numbers as well as detail configurations of the multi-quarks. As QCD does not forbid such multi-quark states, the width is the key to understand why we do not see many ‘‘exotic’’ hadrons in nature. Their possibility should be further pursued both experimentally and theoretically. References [1] [2] [3] [4]
R.J. Jaffe, Phys. Rev. D 15 267 (1977) 281. H.-X. Chen, A. Hosaka, S. -L. Zhu, hep-ph/0609163. Z.-G. Wang, W.-M. Yang, S.-L. Wan, J. Phys. G 31 (2005) 971. H.-J. Lee, Eur. Phys. J. A 30 (2006) 423.
Physica E 40 (2007) 410–413 www.elsevier.com/locate/physe
Exotic quark structure of L(1405) and scalar nonet in QCD sum rule T. Nakamuraa,, J. Sugiyamaa, N. Ishiib, T. Nishikawaa, M. Okaa a
Department of Physics, H-27, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan b Center for Computational Science, University of Tsukuba,Tsukuba, Ibaraki 305-8571, Japan Available online 23 June 2007
Abstract Mixing of multi-quark components in the non-singlet scalar mesons and the flavor singlet L(1405) baryon with negative parity are studied in the QCD sum rule. We propose a formulation to evaluate cross-correlators of q¯q(qqq) and qq¯qq¯ ðqqqq¯qÞ operators and to define mixing of different Fock states in the sum rule. It is applied to the non-singlet scalar mesons and L(1405) baryon. We find that the multi-quark operators predict lower masses than the q¯q(qqq) operators and that the multi-quark states are dominant in the lowest mass states. r 2007 Elsevier B.V. All rights reserved. PACS: 12.38.Lg; 12.39.Mk Keywords: Exotic hadron; QCD sum rule
1. Introduction Recently, the spectroscopy of old (ordinary) hadrons has been re-examined and various possibilities of multiquark components of hadrons have been pointed out. We propose a new analysis of L(1405) and scalar nonet mesons as multi-quark states in QCD sum rule (QCDSR). We, in particular, consider possible mixing of different Fock states. Unnatural spectrum of the light scalar mesons indicates that they are tetra-quark states [1]. The lightest negative-parity baryon L(1405) remains as a mystery in baryon spectroscopy and attracts a lot of attention in the context of strong anti-K-nucleon interactions. We consider here two possible structures of these hadrons: (1) p-wave q¯q or qqq states, and (2) s-wave tetra- or penta-quark states. 2. Formalism In the meson case, the sum rule is obtained by expressing a two point function Z 2 Pðp Þ ¼ i d4 x eipx h0jJðxÞJ y ð0Þj0i, (1)
in two ways. One of them is based on the operator product expansion (OPE), where Eq. (1) is calculated in deep Euclidean region, p2 ¼ 1, and is described in terms of the QCD parameters. The other one is based on a phenomenological parametrization of the spectral function. The spectral function at the physical region (p2 40) is assumed to have a sharp peak resonance at p2 ¼ m2 and continuum at p2 4sth . The sum rule is obtained by matching two expressions, so that the mass of the resonance, m, and the other phenomenological parameters can be expressed by the QCD parameters. Two expressions of the correlators are connected by the dispersion relation. To simplify the sum rule, we approximate the continuum spectrum by the same form as that of the OPE calculation. In order to 2 2 suppress large p2 region by the factor of ep =M , and thus the effects of the continuum, we further apply the Borel transformation. In the baryon case, a similar formalism is used. We employ the following local operators for a 0 and flavor singlet baryon L : J 2 ¼ ð¯ua d a Þ, J 4 ¼ abc dec ðd Ta Cg5 sb Þð¯sd g5 C u¯ Te Þ, J 3 ¼ abc ½ðuTa Cg5 d b Þsc ðuTa Cd b Þg5 sc ðuTa Cg5 gm d b Þgm sc ,
Corresponding author.
E-mail addresses: [email protected], [email protected] (T. Nakamura). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.06.043
J 5 ¼ abc def cfg ðd Ta Cg5 sb ÞðsTd Cg5 ue Þg5 C¯sTg þ ðflavor cyclicÞ,
ð2Þ
ARTICLE IN PRESS T. Nakamura et al. / Physica E 40 (2007) 410–413
411
where a; b; . . . represent colors and C ¼ ig2 g0 . Similarly, the 2-quark and 4-quark operators for K 0 are given by the SU f ð3Þ rotation from the fields for a0 . The OPE is truncated after the dimension-6 terms for the J 2 –J 2 correlator, P22 (or the J 3 –J 3 correlator, P33 ). In order to deal with the same order of power corrections, we expand the cross-correlator, J 2 –J 4 , P24 (P35 ) up to dim-9 and P44 (P55 ) up to dim-12. As usual, high dimension operators are evaluated by the vacuum saturation approximation. We consider the diagrams containing the q¯q annihilation because it is necessary for dealing with the mixing of different Fock states. The quark pair annihilations are substituted for h¯qqi or h¯qgs s Gqi. Because the interpolating fields are under the normal ordering, the perturbative part of the q¯q annihilation must disappear. Because the OPE is represented as a polynomial in x, only the zeroth order term survives in the x ! 0 limit. The main goal of this study is to determine whether the scalar mesons and L are dominated by multi-quark states. It turns out that such a mixing is not easily quantified. It would be natural to consider the strengths of the couplings of the 2-quark (3-quark) and 4-quark (5-quark) operators to the physical state and then evaluate the mixing angle. However, such a procedure is largely dependent on the definition and normalization of the local operators. Indeed, a numerical factor can be easily hidden in the local operators and thus the magnitudes of the coupling strengths are ambiguous. Here, we propose two ways to ‘‘define’’ the ratio of the Fock space probability in the mixing of q¯q and qq¯qq¯ for the scalar (a0 ) meson or similarly for L . In the first approach, we define local operators ‘‘normalized’’ in the context of a full 4-quark operator J 4 in Eq. (2). In fact, the 4-quark operator J 4 contains q¯q component, which is obtained by contracting q¯ q pair into the quark condensate,
Therefore, it does not necessarily have a direct relation to the mixing parameters employed in the quark models. In order to define a mixing angle more appropriately to the quark models, one must determine the normalization of the operators using quark model wave functions. To this end, we employ the bag model and compute the matrix elements,
J 4 ðxÞ ¼ J 04 ðxÞ þ 16h¯ssiJ 2 ðxÞ . |fflfflfflfflfflffl{zfflfflfflfflfflffl}
3. Results and conclusion
(3)
J 02
We regard J 02 and J 04 as ‘‘normalized’’ 2-quark and 4-quark fields, respectively. The quark condensate supplies the dimension of the normalization. Using J 02 ðxÞ and J 04 ðxÞ, we define the mixing parameter, y, so that J a ðxÞ ¼ cos yJ 02 ðxÞ þ sin yJ 04 ðxÞ couples to the physical state most strongly h0jJ a ðxÞja0 i ¼ lfðxÞ,
(4)
where fðxÞ denotes the wave function of the center of mass motion of a0 (i.e., a plane wave for a momentum eigenstate). The mixing parameter can be evaluated from correlation functions with an assumption that the poles are at the same position. This definition of the mixing is model independent, but it depends on the choice of the local operators.
h0jJ 2 ð0Þjð¯qqÞP0 ibag ¼ l2 fð0Þ, h0jJ 4 ð0Þjðqq¯qq¯ ÞS0 ibag ¼ l4 fð0Þ. Now, assuming the bag model states (with definite number of quarks) are normalized properly, one can use l2 and l4 for the normalizations of the operators J 2 and J 4 , respectively. In calculating the mixing parameter, one needs only the ratio of l2 and l4 , which is given by l4 =l2 0:24iR32 =R64 . R is the bag radius and gives the dimensional scale of the normalizations of the two operators that have different dimensions. We assume here that the bag radius of the q¯q state is the same as that of qq¯qq¯ state. Thus, the physical state for a0 is given by the mixing of the two states, ja0 i ¼ i cos yjð¯qqÞP0 i þ sin yjðqq¯qq¯ ÞS0 i ¼ cos yja0 ð2qÞi þ sin yja0 ð4qÞi. Note here that the factor i is necessary to keep the phases of the 2- and 4-quark states in accord so that the mixing parameter can be defined as a real parameter. In the QCDSR, the mixing parameter with the bagmodel normalization can be calculated as R Im d4 x eipx h0jJ 04 ðxÞðJ 04 Þy ð0Þj0i l4 2 2 ¼ (5) R 4 l tan y. 2 Im d x eipx h0jJ 2 ðxÞJ y2 ð0Þj0i This definition is model dependent, but it gives a direct interpretation associated with the quark model.
We calculate the masses of a0 in the case of pure 2-quark and pure 4-quark. The values of QCD parameters are taken as ms ¼ 0:12 GeV, h¯qqi ¼ ð0:23 GeVÞ3 , h¯ssi ¼ 0:8 h¯qqi, h¯qgs s Gqi=h¯qqi ¼ 0:8 GeV2 , has p1 G 2 i ¼ ð0:33 GeVÞ4 and as ¼ 0:4. We find that the Borel stability is fairly good. The positions of the poles in the 2-quark state and the 4-quark states are close to each other. The 4-quark state is slightly light. We plot the mass of the mixed a0 and the mixing parameter defined by the first normalization method in the case of various threshold parameter, sth , in Fig. 1. We see the mixing parameter is almost independent of the threshold and the Borel mass, M. The weak M dependence of the mixing parameter, which is dimensionless, is attributed to cancellation of the p-dependences (M-dependences) of the 2-quark and 4-quark correlators. The mixing parameter is about 701, or in other words, the 4-quark component
ARTICLE IN PRESS T. Nakamura et al. / Physica E 40 (2007) 410–413
occupies 90% of a0 . The predicted mass extracted from the mixed operator, J a ðxÞ, is about 0.9–1.1 GeV, which agrees well with experiment. The mixing parameters from the normalized operators by the bag model are estimated for various bag radii. We take the central value as R ¼ 4:8 GeV1 , which is the bag radius determined for the q¯ q component of a0 . The mixing parameter for R ¼ 5:2 GeV1 is almost the same as the one in the first normalization method. We find that the dominant components of a0 , about more than 70%, are the 4-quark state for R44:4 GeV1 and for the first normalization method. We examine how well these sum rules work. We calculate the pole dominance signature defined by B POPE ðp2 Þyðsth p2 Þ . (6) B½POPE ðp2 Þ We set the Borel window at the region where the pole dominance is more than 30%. This constraint is weaker than the standard criterion, but the results are not sensitive to the choice of this value. One finds that the pole dominance for the pure 4-quark correlator is less than the pure 2-quark correlator. The reason is that the OPE of the 4-quark correlator contains higher powers of p2 and grows rapidly for large p2 . The sum rule for the mixed operator shows similar results. We set the Borel window Mo1:2 GeV and conclude that the 4-quark component is dominant in a0 . The results of K 0 are obtained by the SU f ð3Þ rotation from the results for a0 . Comparing to a0 , the mass extracted from the 2-quark correlator for K 0 is heavier, while that of the 4-quark components lies at about the same mass as a0 . The mass of the experimental values for K 0 has been reported at about 0.84 GeV, and its width seems very wide, about 600 MeV. The scalar nonet 1.4
90.0 (100%)
1.3
67.5 (85%)
1.2
45.0 (50%)
1.1
22.5 (15%)
spectrum from our sum rule for 4-quark components is more consistent with the experimental value than that for 2-quark components. The position of the pole of K 0 for the 2-quark operators starts splitting at low M from that for the 4-quark operators. We, however, evaluate the mixing parameter by assuming that the positions of the poles are at the same position. One finds that the 4-quark component is dominant, occupying about 70–90% of K 0 . These results for K 0 resemble those for a0 . The behavior of the pole dominance for K 0 is also similar to that for a0 . Next we calculate the mass of L by the pure 3-quark and 5-quark operators. It is found that the low-lying spectrum is dominated by the negative parity state. We find the pole positions in 3-quark state and 5-quark state are close. Therefore, we estimate the mass of the mixed L and the mixing parameter defined by the first normalization method in the case of various threshold parameters, sth , in Fig. 2. We see the mixing parameter is about 701, or the 5quark component occupies 90% of L . In the second normalization, we take the central value as R ¼ 5:5 GeV1 , which is the bag radius determined for the qqq component. Then we find that the dominant component of L , about more than 90%, is the 5-quark state. We set the Borel window in the region where the pole dominance is more than 30%. We find the Borel window Mo1:4 GeV and conclude that the 5-quark component is dominant in L . The obtained mass is consistent with the observed baryon, L(1405). We summarize the results in this work. A formulation is proposed to take into account mixing of different Fock states in the QCDSR. We apply the formulation to the scalar mesons and the flavor singlet L . Our sum rules indicate that the 4-quark component occupy 70–90% of the scalar mesons. We find that K 0 is lighter than a0 , which is consistent with the strange-quark counting when these
0.0
(0%)
–22.5 (15%)
0.9 mass (√sth =1.3GeV) mixing parameter (√sth =1.3GeV) mass (√sth =1.4GeV) mixing parameter (√sth =1.4GeV) mass (√sth =1.5GeV) mixing parameter (√sth =1.5GeV)
0.8 0.7 0.6 0.8
1 M [GeV]
–45.0 (50%) –67.5 (85%) –90.0 (100%) 1.2
Fig. 1. The masses of the a0 mixed between 2-quark and 4-quark components are plotted as a function of the Borel mass, M, in the case of pffiffiffiffiffiffi various threshold parameters, sth ¼ 1:3, 1.4 and 1.5 GeV. The mixing parameter of the first normalization method is also plotted and it seems identical in the cases of any threshold parameters.
90.0 (100%) 67.5 (85%)
1.6
45.0 (50%) Mass of Λ* [GeV]
1
mixing parameter [degree]
Mass of a0 [GeV]
1.8
1.4
22.5 (15%)
1.2
0.0
–22.5 (15%)
mass ( √sth =1.6GeV) mixing parameter ( √sth =1.6GeV) mass ( √sth =1.8GeV) mixing parameter ( √sth =1.8GeV) mass ( √sth =2.0GeV) mixing parameter ( √sth =2.0GeV)
1 0.8 0.6 1
1.2
1.4 1.6 M [GeV]
1.8
(0%)
–45.0 (50%)
mixing parameter [degree]
412
–67.5 (85%) –90.0 (100%) 2
Fig. 2. The mass of L , where the 3-quark and 5-quark components are mixed, is plotted as a function of the mass, M, for various threshold pffiffiffiffiffiffi parameters, sth ¼ 1:6, 1.8 and 2.0 GeV. The mixing parameter according to the first normalization method is also plotted. The results for three choices of sth are almost identical.
ARTICLE IN PRESS T. Nakamura et al. / Physica E 40 (2007) 410–413
states are considered as 4-quark states. L(1405) is also dominated by the penta-quark component, whose probability is estimated as more than 90%. There exist several other studies of the 4-quark scalar nonet in the QCDSR. Two of them [2,3] are consistent with our result in terms of reproducing the lighter 4-quark mass than the 2-quark mass. In these analyses, K 0 is predicted to have smaller mass than a0 . We think that the difference of the threshold parameter probably makes K 0 lighter than a0 . Another work [4] claims that no signal is obtained for the 4-quark scalar nonet. We note that their analysis differs from ours in not considering the mixing diagrams, truncating dimension in the OPE, interpolating field and the definition of the coupling strength of ground state. It is important to examine the decay widths and branching ratios of those multi-quark states. The mechan-
413
ism for the fall-apart decay in which the multi-quark hadrons dissociate into two color-singlet hadrons without creating q¯q pairs. The widths associated with the fall-apart processes depend strongly on the quantum numbers as well as detail configurations of the multi-quarks. As QCD does not forbid such multi-quark states, the width is the key to understand why we do not see many ‘‘exotic’’ hadrons in nature. Their possibility should be further pursued both experimentally and theoretically. References [1] [2] [3] [4]
R.J. Jaffe, Phys. Rev. D 15 267 (1977) 281. H.-X. Chen, A. Hosaka, S. -L. Zhu, hep-ph/0609163. Z.-G. Wang, W.-M. Yang, S.-L. Wan, J. Phys. G 31 (2005) 971. H.-J. Lee, Eur. Phys. J. A 30 (2006) 423.